Scalene Triangle - Sides, Area, and Angles

The scalene triangle calculator converts three different side lengths into area, perimeter, semi-perimeter, and the three interior angles in a single pass. A scalene triangle has all three sides of different lengths and all three angles of different sizes, so once the three sides are known the whole shape is described. The tool runs Heron's formula for the area and the law of cosines for the angles.

• • Homework and exam checks: Verify area, perimeter, and angle steps for a geometry problem when only the three side lengths are given.
• • Reverse solving: Start from three measured sides of a real object and recover the area, perimeter, and angles without measuring a height or angle.
• • Layout and cutting estimates: Estimate material takeoffs for fabric, sheet metal, wood, or paper when the piece is a scalene triangle with three measured edges.
• • Quick design sketches: Compare candidate scalene triangle proportions in a draft by entering different side lengths and reading off area and angle changes.

A scalene triangle is the most general triangle. Unlike an isosceles triangle, which has at least two equal sides, or an equilateral triangle, where all three sides are equal, the scalene triangle has no symmetry to lean on. The three side lengths are the only inputs the tool needs.

The three side lengths are first checked against the triangle inequality: any two sides must sum to more than the third. If they do not, no closed triangle can be drawn and the tool returns a clear error. When the inequality holds, the calculator classifies the triangle as acute, right, or obtuse, then runs the area and angle formulas.

If you only need the area, semi-perimeter, and perimeter from three sides, the Scalene Triangle Area Calculator uses Heron's formula without the extra angle and classification step.

The calculator runs Heron's formula to compute the area, then applies the law of cosines once per angle to recover the three interior angles.

• a, b, c: the three side lengths entered by the user, in any consistent length unit
• s (semi-perimeter): (a + b + c) / 2, the half-perimeter that drives Heron's formula
• area: the inside area of the triangle, in the square unit that matches the input length unit
• perimeter: a + b + c, the outside distance around the triangle
• angle A, B, C: the three interior angles, one opposite each side, returned in degrees

Heron's formula only needs the three side lengths. The semi-perimeter s is the perimeter divided by two, and plugging s and the three side differences into the square root gives the area. The same three side lengths feed the law of cosines, which is the companion rule for turning sides into angles.

The largest side decides whether the triangle is acute, right, or obtuse. If the square of the largest side equals the sum of the squares of the other two sides, the triangle is right. If the largest side's square is bigger, the triangle is obtuse. If it is smaller, the triangle is acute.

According to Wolfram MathWorld, Heron's formula gives the area of a triangle with side lengths a, b, c as the square root of s(s-a)(s-b)(s-c), where s is the semi-perimeter (a+b+c)/2.

When the data you have is two sides and an included angle rather than three sides, the Triangle Calculator handles the SAS and SSS paths in one place.

These four ideas are the language you will see in the formula box, the worked example, and the FAQ, so it helps to define them before using the calculator.

If the scalene triangle turns out to be right, the Right Triangle Calculator is the natural follow-up for Pythagorean theorem and angle work on the 90-degree case.

Only the three side lengths are needed. The tool handles the inequality check, the classification, and all the formulas from there.

1. 1 Enter side a: Type the length of the first side of the scalene triangle in your chosen unit, such as meters, feet, inches, or centimeters.
2. 2 Enter side b: Type the length of the second side. Use the same unit you used for side a, since the area will come back in the matching square unit.
3. 3 Enter side c: Type the length of the third side. The three lengths must be different from each other to describe a scalene triangle, and they must satisfy the triangle inequality.
4. 4 Read the area and perimeter: Use the Area and Perimeter results for material counts, coverage estimates, or perimeter-based comparisons.
5. 5 Read the three interior angles: Use the three angle results for layout work, miter cuts, or to verify that the angles sum to 180 degrees.
6. 6 Check the triangle type: Read the triangle type to see whether the scalene triangle is acute, right, or obtuse.

A landscape designer has a triangular flower bed with sides 4 meters, 5 meters, and 6 meters. Entering those three values returns area 9.92 square meters, perimeter 15 meters, semi-perimeter 7.5 meters, angles 41.41, 55.77, and 82.82 degrees, and type acute. The area tells the designer how much soil to buy, the perimeter sets the edging length, and the angles guide the cut of any stone border.

If you only have two sides and the included angle for a scalene-shaped problem, the SAS Triangle Calculator takes that data and returns the same area, perimeter, and angles.

Running the side, area, perimeter, and angle calculations in one place keeps the result consistent and easy to audit.

• • Three outputs in one pass: Area, perimeter, semi-perimeter, three interior angles, and the acute/right/obtuse type come back together, so there is no risk of mixing units or formulas between two separate tools.
• • No height or angle required: Heron's formula only needs the three side lengths, which are usually the easiest measurements to take on a real object.
• • Validates the triangle inequality: The calculator checks the triangle inequality before computing anything, so a bad input fails fast with a clear message instead of returning a meaningless area or angle.
• • Classifies the triangle automatically: The acute, right, or obtuse label is set from the largest side, so the result tells you which kind of scalene triangle you are working with.

When two of the three sides are equal and the scalene assumption no longer holds, the Isosceles Triangle Area Calculator uses the symmetry to skip the law of cosines step.

The calculator runs on compact math, but a few decisions about the inputs and the assumptions affect how to read the result.

• • This calculator is a pure geometry tool. It does not solve for a missing side from two known sides and an angle, which is the SAS Triangle Calculator workflow.
• • The results are geometric estimates only. Real material takeoffs for fabric, sheet metal, or wood need extra allowances for seams, overlap, cutting waste, and edge thickness.

According to Wolfram MathWorld, for a triangle with sides a, b, c opposite to angles A, B, C, the angle opposite a is arccos((b^2 + c^2 - a^2) / (2bc)).

For the limiting case where all three sides are equal, the Equilateral Triangle Area gives a closed-form area and angle result that confirms the scalene result approaches the equilateral limit.